Characterization of eventually periodic modules in the singularity categories

TL;DR

This paper characterizes eventually periodic modules over artin rings within the singularity category, linking their properties to morphisms and exploring invariance under singular equivalences.

Contribution

It provides a new characterization of eventually periodic modules in the singularity category and studies their invariance under certain algebraic equivalences.

Findings

Characterization of eventually periodic modules via morphisms in the singularity category

Invariance of eventual periodicity under singular equivalence of Morita type with level

Classification of eventually periodic Nakayama algebras over algebraically closed fields

Abstract

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.